###### Oscillation and Periodicity of a Second Order Impulsive Delay Differential Equation with a Piecewise Constant Argument

References [1] M. Akhmet: Nonlinear hybrid continuous/discrete-time models.. Springer Science & Business Media (2011). [2] H. Bereketoglu, G.S. Oztepe: Convergence of the solution of an impulsive differential equation with piecewise constant arguments. Miskolc Math. Notes 14 (2013) 801-815. [3] H. Bereketoglu, G.S. Oztepe: Asymptotic constancy for impulsive differential equations with piecewise constant argument. Bull. Math. Soc. Sci. Math. Roumanie Tome 57 (2014) 181-192. [4] H. Bereketoglu, G

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Generalized Jacobsthal numbers and restricted *k*-ary words

## Abstract

We consider a generalization of the problem of counting ternary words of a given length which was recently investigated by Koshy and Grimaldi [10]. In particular, we use finite automata and ordinary generating functions in deriving a *k*-ary generalization. This approach allows us to obtain a general setting in which to study this problem over a *k*-ary language. The corresponding class of *n*-letter *k*-ary words is seen to be equinumerous with the closed walks of length *n* − 1 on the complete graph for *k* vertices as well as a restricted subset of colored square-and-domino tilings of the same length. A further polynomial extension of the *k*-ary case is introduced and its basic properties deduced. As a consequence, one obtains some apparently new binomial-type identities via a combinatorial argument.

###### On the notion of Jacobi fields in constrained calculus of variations

## Abstract

In variational calculus, the minimality of a given functional under arbitrary deformations with fixed end-points is established through an analysis of the so called second variation. In this paper, the argument is examined in the context of constrained variational calculus, assuming piecewise differentiable extremals, commonly referred to as extremaloids. The approach relies on the existence of a fully covariant representation of the second variation of the action functional, based on a family of local gauge transformations of the original Lagrangian and on a set of scalar attributes of the extremaloid, called the corners' strengths [16]. In dis- cussing the positivity of the second variation, a relevant role is played by the Jacobi fields, defined as infinitesimal generators of 1-parameter groups of diffeomorphisms preserving the extremaloids. Along a piecewise differentiable extremal, these fields are generally discontinuous across the corners. A thorough analysis of this point is presented. An alternative characterization of the Jacobi fields as solutions of a suitable accessory variational problem is established.

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Approach of *q*-Derivative Operators to Terminating *q*-Series Formulae

–271. [28] Z. Liu: An expansion formula for q -series and applications. Ramanujan J. 6 (4) (2002) 429–447. [29] B.M. Minton: Generalized hypergeometric function of unit argument. J. Math. Phys. 11 (4) (1970) 1375–1376. [30] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/17.7 [31] I.J. Slater: Generalized Hypergeometric Functions . Cambridge University Press, Cambridge (1966). [32] A. Verma, V.K. Jain: Some summation formulae for nonterminating basic hypergeometric series. SIAM J. Math. Anal. 16 (3) (1985) 647

###### Integrals of logarithmic and hypergeometric functions

References [1] V. Adamchik, H. M. Srivastava: Some series of the zeta and related functions. Analysis 18 (2) (1998) 131-144. [2] J. M. Borwein, I. J. Zucker, J. Boersma: The evaluation of character Euler double sums. Ramanujan J. 15 (2008) 377-405. [3] J. Choi: Log-Sine and Log-Cosine Integrals. Honam Mathematical J 35 (2) (2013) 137-146. [4] J. Choi, D. Cvijović: Values of the polygamma functions at rational arguments. J. Phys. A: Math. Theor. 40 (50) (2007) 15019{15028. Corrigendum, ibidem, 43

###### Stability and synchronization of delayed fractional-order projection neural network with piecewise constant argument of mixed type

## Abstract

Projection equations arise in several optimization problems and possess significant applications in many areas of science and engineering. In this paper, we propose a fractional-order projection neural network to solve quadratic programming problems. We study stability and synchronization for a class of delayed projection neural networks of mixed type via impulsive control. Using concepts of fractional calculus, we investigate the existence of solution and study its global asymptotic stability. Moreover, we propose an effective impulsive control scheme to achieve synchronization for the system. We demonstrate the validity and transient behaviour of the proposed neural network with the help of suitable examples.

###### Uniform Euler approximation of solutions of fractional-order delayed cellular neural network on bounded intervals

-301. [26] Chiu K.S.; Pinto, M.; Jeng J.C. Existence and global convergence of periodic solutions in recurrent neural network models with a general piecewise alternately advanced and retarded argument. Acta Applicandae Mathematicae 136.1 (2015): 193-216. [27] Pinto, M. Asymptotic equivalence of nonlinear and quasi linear differential equations with piecewise constant arguments. Mathematical and Computer Modelling 49 (2009): 451-458. [28] Pinto, M.; Robledo G. Controllability and observability for a linear time varying system with piecewise

###### On fixed-point theorems in synthetic computability

. [9] S. C. Kleene. On notation for ordinal numbers. Journal of Symbolic Logic, 3(4):150-155, 1938. [10] S. C. Kleene. Introduction to Metamathematics. North-Holland, 1952. [11] B. Knaster. Un théoréme sur les fonctions d’ensembles. Annales Polonici Mathematici, 6:133-134, 1928. [12] J. Lambek and P. J. Scott. Introduction to higher order categorical logic, volume 7 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1986. [13] F. W. Lawvere. Diagonal arguments and

###### Algebraic Kan extensions along morphisms of internal algebra classifiers

References [1] M. Batanin. Monoidal globular categories as a natural environment for the theory of weak n-categories. Advances in Mathematics, 136:39{103, 1998. [2] M. Batanin. The Eckmann-Hilton argument and higher operads. Advances in Mathematics, 217:334{385, 2008. [3] M. Batanin and C. Berger. Homotopy theory for algebras over polynomial monads. ArXiv:1305.0086. [4] M. Batanin, J. Kock and M. Weber. Feynman categories are operads, regular patterns are substitudes. In preparation. [5] R. Blackwell, G. M. Kelly and A. J. Power. Two-dimensional monad

###### New families of alternating harmonic number sums

References [1] J. M. Borwein, I. J. Zucker and J. Boersma. The evaluation of character Euler double sums. Ramanujan J. 15 (2008), 377-405. [2] J. Choi and D. Cvijovic. Values of the polygamma functions at rational arguments. J. Phys. A: Math. Theor. 40 (2007), 15019-15028; Corrigendum, ibidem, 43 (2010), 239801 (1 p). [3] J. Choi. Finite summation formulas involving binomial coefficients, harmonic numbers and generalized harmonic numbers. J. Inequal. Appl. 49 (2013), 11p. [4] J. Choi, and H. M. Srivastava. Some summation formulas involving harmonic